You must have purchased the Going Rogue expansion to have full access to Morality Missions and Hero Merits. If you don't have Going Rogue, you can still play Tips, but they won't count toward anything except XP, and you won't get any Merits.

Timing

Every day in real time, you can get Public Fame points for playing up to 5 Tips on each character. Once you've done 5 Tips, you won't get any more credit for playing tips on that character. (That's usually a good time to switch to another alternate character and play 5 more tips on him/her!).

Credit Accumulation and Finding Morality Missions

Once you've accumulated credit for playing 10 tips, check to see if you have a "Morality" mission in your Tips tab. If you don't, and you're full on Tips (you can only have 3 tips in your Tips tab at one time), you'll have to play another of your available Tips and see if your Morality mission will appear while you're playing it.

Spending Your Rewards

Once you've played your Morality mission, you'll get either 50 reward merits (if it's your first completed Morality mission) or a Hero Merit. Hero Merits can be traded in Fort Trident; the following table shows what rewards can be had for various numbers of Hero Merits spent.

Reward Merits can be traded in for items at any Merit Vendor. They are located in most normal Hero Zones (not Hazard zones or the Shadow Shard). You can buy anything from large Inspirations to a random rare Recipe at these vendors.